We have in this work encountered up to now two types of perturbative calculations, one is based on the isospin asymmetry \(\delta \)  and the other is based on the expansion around some reference density \(\rho _{\mathrm {f}}\)  through the quantity \(\theta _{\mathrm {f}}\)  (see (REF )).
These two expansions both give us useful/informative information on the EOS of ANM, e.g., as we discussed in the last section, the \(\delta ^2\) -expansion generally leads to the parabolic approximation, i.e., the fourth-order symmetry energy and beyond are smaller than the quadratic symmetry energy.
In addition, via the HVH theorem, a novel expansion related to the \(\delta \)  was developed, leading to the useful expressions for the symmetry energy as well as its slope parameter in terms of the single-nucleon potentials, with the latter being extractable from experiments [1]}, [2]}.
If one treats the dimension \(d\)  as a continuous variable, one can also develop corresponding perturbation theories based on certain (fixed) dimension \(d_{\mathrm {f}}\)  by introducing the (apparently) small quantity \(\epsilon =d-d_{\mathrm {f}}\) . This approach is often called the \(\epsilon \) -expansion theory. Some problems are well defined and easier to analyze in certain dimensions, while they are difficult or even impossible to study in other dimensions.
The \(\epsilon \) -expansion was invented in the 1970's to deal with the second-order phase transition problems and related critical phenomena with impressive successes [3]}, [4]}, [5]}. A recent successful example of the \(\epsilon \) -expansion is in studying the EOS of a unitary Fermi gas. The latter could simply be written as \(\xi \cdot (3E_{\mathrm {F}}/5)\)  where \(E_{\mathrm {F}}=k_{\mathrm {F}}^2/2M\)  is the Fermi energy with \(M\)  being the mass of the atoms.
Here the universal dimensionless quantity \(\xi \)  characterizes the interacting nature of the unitary gas, and it is often called the Bertsch's parameter [6]}, [7]}.
Due to the infinite scattering length \(a_{\mathrm {s}}\)  near the unitary limit, there are currently no self-consistent theories on the BEC-BCS crossover physics [8]}, [9]}, [10]} and thus lacks direct theories on the Bertsch's parameter.
The effective theories based on the \(\epsilon \) -expansion to order \(\epsilon ^{5/2}\ln \epsilon \)  [11]} and later to higher orders in \(\epsilon \)  successfully predicted the value of \(\xi \approx 0.367\)  [12]}, which was found to be very close to the experimental result of about 0.376 [13]}.
